Boundedness and continuity of additive and convex functionals (Q915068)

Let X be a real Baire topological vector space and let A(X), B(X) be classes of sets \(T\subset X\) defined as follows: \(T\in A(X)\) iff for any open convex set \(D\supset T\), every Jensen convex functional defined on D and bounded from above on T, is continuous; \(T\in B(X)\) iff every additive functional on X, bounded from above on T, is continuous. The main result, \(A(X)=B(X)\), generalizes an earlier result by \textit{M. E. Kuczma} [Fund. Math. 66, 383-392 (1970; Zbl 0194.360)].